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Properties

Label 21.21.2642818916...7744.1
Degree $21$
Signature $[21, 0]$
Discriminant $2^{12}\cdot 7^{14}\cdot 13^{2}\cdot 83^{2}\cdot 757^{2}\cdot 3375499^{2}\cdot 3537627827^{2}$
Root discriminant $675.50$
Ramified primes $2, 7, 13, 83, 757, 3375499, 3537627827$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 21T153

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-11087, 118116, -188374, -1250551, 5357467, -7366310, 1467637, 5830445, -4536095, -1003737, 2137623, -238467, -468489, 116749, 56171, -18845, -3786, 1525, 135, -62, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 2*x^20 - 62*x^19 + 135*x^18 + 1525*x^17 - 3786*x^16 - 18845*x^15 + 56171*x^14 + 116749*x^13 - 468489*x^12 - 238467*x^11 + 2137623*x^10 - 1003737*x^9 - 4536095*x^8 + 5830445*x^7 + 1467637*x^6 - 7366310*x^5 + 5357467*x^4 - 1250551*x^3 - 188374*x^2 + 118116*x - 11087)
 
gp: K = bnfinit(x^21 - 2*x^20 - 62*x^19 + 135*x^18 + 1525*x^17 - 3786*x^16 - 18845*x^15 + 56171*x^14 + 116749*x^13 - 468489*x^12 - 238467*x^11 + 2137623*x^10 - 1003737*x^9 - 4536095*x^8 + 5830445*x^7 + 1467637*x^6 - 7366310*x^5 + 5357467*x^4 - 1250551*x^3 - 188374*x^2 + 118116*x - 11087, 1)
 

Normalized defining polynomial

\( x^{21} - 2 x^{20} - 62 x^{19} + 135 x^{18} + 1525 x^{17} - 3786 x^{16} - 18845 x^{15} + 56171 x^{14} + 116749 x^{13} - 468489 x^{12} - 238467 x^{11} + 2137623 x^{10} - 1003737 x^{9} - 4536095 x^{8} + 5830445 x^{7} + 1467637 x^{6} - 7366310 x^{5} + 5357467 x^{4} - 1250551 x^{3} - 188374 x^{2} + 118116 x - 11087 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $21$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[21, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(264281891642047276904710624389705471956890680461868233887744=2^{12}\cdot 7^{14}\cdot 13^{2}\cdot 83^{2}\cdot 757^{2}\cdot 3375499^{2}\cdot 3537627827^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $675.50$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 7, 13, 83, 757, 3375499, 3537627827$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{10} - \frac{1}{4} a^{8} + \frac{1}{4} a^{4} + \frac{1}{4} a^{2} - \frac{1}{4}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{11} - \frac{1}{4} a^{9} + \frac{1}{4} a^{5} + \frac{1}{4} a^{3} - \frac{1}{4} a$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{8} - \frac{1}{4} a^{6} + \frac{1}{4}$, $\frac{1}{4} a^{15} - \frac{1}{4} a^{9} - \frac{1}{4} a^{7} + \frac{1}{4} a$, $\frac{1}{4} a^{16} - \frac{1}{4} a^{10} - \frac{1}{4} a^{8} + \frac{1}{4} a^{2}$, $\frac{1}{4} a^{17} - \frac{1}{4} a^{11} - \frac{1}{4} a^{9} + \frac{1}{4} a^{3}$, $\frac{1}{104} a^{18} + \frac{3}{104} a^{17} - \frac{3}{26} a^{16} - \frac{7}{104} a^{15} - \frac{11}{104} a^{14} + \frac{1}{26} a^{13} - \frac{1}{8} a^{12} - \frac{15}{104} a^{11} - \frac{21}{104} a^{10} + \frac{3}{26} a^{9} + \frac{3}{104} a^{8} - \frac{25}{104} a^{7} - \frac{9}{104} a^{6} - \frac{1}{26} a^{5} + \frac{21}{104} a^{4} - \frac{1}{104} a^{3} + \frac{4}{13} a^{2} - \frac{47}{104} a - \frac{23}{104}$, $\frac{1}{11913285904} a^{19} + \frac{6569663}{2978321476} a^{18} - \frac{1436873507}{11913285904} a^{17} + \frac{76696817}{916406608} a^{16} - \frac{102533851}{5956642952} a^{15} - \frac{1473191351}{11913285904} a^{14} + \frac{807794847}{11913285904} a^{13} - \frac{161257379}{1489160738} a^{12} + \frac{892934161}{5956642952} a^{11} - \frac{217847921}{11913285904} a^{10} - \frac{1343593203}{11913285904} a^{9} + \frac{1465114259}{5956642952} a^{8} + \frac{1442981201}{5956642952} a^{7} - \frac{2027679825}{11913285904} a^{6} + \frac{3978692893}{11913285904} a^{5} + \frac{194934559}{744580369} a^{4} + \frac{5450213729}{11913285904} a^{3} - \frac{2778202003}{11913285904} a^{2} + \frac{204306745}{5956642952} a + \frac{4261716701}{11913285904}$, $\frac{1}{17740797628805637152} a^{20} + \frac{13139325}{17740797628805637152} a^{19} + \frac{172641887734213}{17740797628805637152} a^{18} + \frac{1005659878596774309}{8870398814402818576} a^{17} - \frac{482849170140575805}{17740797628805637152} a^{16} + \frac{113077965640195427}{17740797628805637152} a^{15} - \frac{157296768005649347}{2217599703600704644} a^{14} - \frac{1042476644789527845}{17740797628805637152} a^{13} + \frac{1064139306321120973}{8870398814402818576} a^{12} + \frac{3486636319712656029}{17740797628805637152} a^{11} + \frac{9078438226867899}{85292296292334794} a^{10} + \frac{1745927049666941655}{17740797628805637152} a^{9} - \frac{511366433825903663}{2217599703600704644} a^{8} + \frac{2424719701373491337}{17740797628805637152} a^{7} - \frac{254721384066592237}{4435199407201409288} a^{6} - \frac{2669112422030903167}{17740797628805637152} a^{5} + \frac{479752482803855281}{17740797628805637152} a^{4} + \frac{9374885010759509}{8870398814402818576} a^{3} - \frac{7097183688470159489}{17740797628805637152} a^{2} - \frac{7524152246707426173}{17740797628805637152} a - \frac{7734580581847419439}{17740797628805637152}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $20$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 197138673304000000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

21T153:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 48009024000
The 267 conjugacy class representatives for t21n153 are not computed
Character table for t21n153 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\)

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 45 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.6.0.1}{6} }{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ $21$ R $21$ R $15{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ ${\href{/LocalNumberField/29.7.0.1}{7} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{3}$ $21$ $15{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.7.0.1}{7} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{5}$ $21$ ${\href{/LocalNumberField/53.9.0.1}{9} }{,}\,{\href{/LocalNumberField/53.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.6.0.1$x^{6} - x + 1$$1$$6$$0$$C_6$$[\ ]^{6}$
2.12.12.31$x^{12} + 4 x^{11} - 6 x^{10} + 8 x^{9} - 4 x^{8} + 8 x^{7} - 4 x^{6} + 4 x^{5} - 4 x^{4} + 8 x + 8$$4$$3$$12$12T205$[4/3, 4/3, 4/3, 4/3, 4/3, 4/3]_{3}^{6}$
$7$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.12.8.1$x^{12} - 63 x^{9} + 637 x^{6} + 6174 x^{3} + 300125$$3$$4$$8$$C_{12}$$[\ ]_{3}^{4}$
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.3.0.1$x^{3} - 2 x + 6$$1$$3$$0$$C_3$$[\ ]^{3}$
13.3.2.3$x^{3} - 52$$3$$1$$2$$C_3$$[\ ]_{3}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.5.0.1$x^{5} - 2 x + 6$$1$$5$$0$$C_5$$[\ ]^{5}$
83Data not computed
757Data not computed
3375499Data not computed
3537627827Data not computed